Connected subsets of R are intervals

Connected subsets of $\mathbb{R}$ are intervals: View $\mathbb{R}$ with its usual topology (for example, the one induced by the standard metric). If $E\subseteq \mathbb{R}$ is connected , then $E$ is an interval : whenever $a,b\in E$ with $a<b$ and $c\in\mathbb{R}$ satisfies $a<c<b$, one has $c\in E$. Conversely, every interval in $\mathbb{R}$ is connected.

This classification is frequently combined with continuous images preserve connectedness to identify images of connected sets under real-valued continuous functions.